Geneva CUSD 304 Content-Area Curriculum Frameworks Grades 6-12 Mathematics

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1 Geneva CUSD 304 Content-Area Curriculum Frameworks Grades 6-12 Mathematics Mission Statement The study of mathematics can be an exciting and interesting challenge. Yet, the major reason to become proficient in this discipline revolves around the use of mathematics as a tool to solve problems from the areas of science, business, engineering, economics, and all other areas which involve data collection and analysis. The mathematics program is designed to establish connections between the key concepts of math and the applications. Theoretical structure of mathematics - Students will demonstrate an understanding of the theoretical foundations of mathematics. Thought processes (intuition, deduction, induction) Logical arguments (two column, narrative, flow chart proofs) Structure of axiomatic systems (Euclidian geometry, real and complex number systems) Fundamental concepts (functions, sets, limits, infinity, infinitesimals, statistics, probability) Problem solving - Students will formulate problem solving strategies. Establishment of relationships (numerical, geometric, pictorial, graphic, symbolic) Recognition, collection, and analysis of pertinent data Development and evaluation of methods and algorithms Validation of results (estimation, approximation, reasonableness) Mechanics of mathematics - Students will symbolically manipulate mathematical expressions and statements. Performance of operations and computational processes (arithmetic, algebraic, graphic) Illustration of solution processes for equations and inequalities Calculation using electronic devices (scientific and graphing) Appropriate use of emerging technology - Students will use technology to improve and extend their understanding of mathematics. Calculators at appropriate levels (scientific and graphing) Computer software (graphic, spread sheets, data bases, Calculus and Calculus AP Frameworks.doc Page 1 of 16

2 symbolic manipulators, simulations) Information management systems (compact disk, telecommunication, internet, video disk) All students will experience an evolving curriculum designed to be a rich tapestry of traditional mathematics skills intertwined with problem solving, graphical analysis, measurement, probability, and statistics. The use of manipulatives, calculators (scientific and graphing), computers, writing assignments, and cooperative learning activities will all be designed to achieve this mission. Course Sequence (Grades 6-12) 6 th Grade Mathematics 7 th Grade Mathematics Pre- Algebra Integrated Mathematics I, II Algebra I (4 semesters) A and B Algebra I Geometry /Concepts and Applications Geometry (Regular & Honors) Algebra II (Regular & Honors) Algebra II 1/3-3/3 Pre-Calculus (Regular & Honors) Trigonometry (Regular & Honors) Calculus AP Calculus AP Statistics Calculus and Calculus AP Frameworks.doc Page 2 of 16

3 Course Framework Course Title Grade Level Semesters Prerequisite Course Description District-approved Materials and/or Resources Calculus 11, 12 2 Pre-Calculus and Trigonometry This is an introductory course in differential and integral calculus. The curriculum begins with a brief review of pre-calculus topics followed by limit theory, differentials and derivatives, basic integration techniques, and their applications. This course is intended for students who want an introduction to basic calculus but who are not planning on earning advanced placement credit in university-level calculus. This course does not receive a weighted grade. Calculus John Wiley & Sons Anton, Bivens, & Davis ISBN Course Title Grade Level Semesters Prerequisite Course Description District-approved Materials and/or Resources AP Calculus (AB) 11, 12 2 Department Approval This course is the final course of the honors sequence. The curriculum includes limits and continuity, differentials and derivatives, integration, and their applications. It follows the Advanced Placement AB Calculus curriculum as established by the College Board. This course is intended for students who have been very successful in their pre-calculus courses and are ready to complete a university- level course in introductory differential and integral calculus. Students taking this course are expected to take the AP exam in the Spring. Calculus John Wiley & Sons Anton, Bivens, & Davis ISBN Calculus and Calculus AP Frameworks.doc Page 3 of 16

4 Unit Frameworks Unit of Study: Pre-Calculus & Limits Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural College Board Goal Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. The should understand the connections among these representations. Students should be able to communicate mathematics both orally and in well written sentences and should be able to explain solutions to problems. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to solve problems, experiment, interpret results, and verify conclusions. 1. Pre-Calculus & Limits 1.1. Slope as a measurement of directionality Linear equations Angle of inclination 1.2. Solution of linear and nonlinear inequalities Solution of absolute value equations and inequalities Calculator as a tool Intervals 1.3. Distance Equations of circles 1.4. Functions Multiple representations - graphically, analytically, numerically, verbally Domains and ranges Types of domains natural, geometric, imposed Locally defined functions Piecewise defined Graphs on the calculator Graphs - analysis by graphing calculator Domains and ranges Calculus and Calculus AP Frameworks.doc Page 4 of 16

5 Asymptotes Family of curves Composites Decomposition Transformations 1.5. Definitions of Limits Graphically Numerically Use of the table function on the calculator Rigorous approach - ε-δ, ε-n and N-δ proofs (AP) 1.6. Evaluation of Limits 1.7. Continuity 1.8. Limits and continuity of trigonometric functions Notes on Unit 1 The students come in to this class with high level of pre-calculus skills. They generally are able to use the graphing calculator to graph functions, solve equations, etc. so very little direct instruction is required. Limits are the first significantly new concept for them. Limits are introduced numerically and graphically, using the graphing calculator, and then formalized with rigorous definitions (AP). This unit is also where students begin to hone their skills on writing mathematics in a structured manner. Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes Multiple Choice Unit Test Free Response Unit Test AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 5 of 16

6 Unit Frameworks Unit of Study: The Derivative Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural College Board Goal Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should use derivatives to solve a variety of problems. Students should be able to use technology to solve problems, experiment, interpret results, and verify conclusions. Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. 2. The Derivative 2.1. Instantaneous rate of change as a limit 2.2. Definition of the Derivative 2.3. Techniques of differentiation The Rules of operations Trigonometric Chain Rule Use of the calculator to verify analytical differentiation 2.4. Implicit functions and differentiation 2.5. Higher order derivatives 2.6. Local linearity 2.7. Differentials Linear approximation 2.8. Related rates 2.9. Rectilinear Motion I Notes on Unit 2 This is when students introduced to the concept of instantaneous and infinitesimal change. Related rates present students with mathematical modeling. Much of the emphasis of this unit is on the mechanics of differentiation. Local linearity introduces the students to numerical differentiation on a calculator. Writing mathematically correct, sequential explanations is Calculus and Calculus AP Frameworks.doc Page 6 of 16

7 emphasized. Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes Multiple Choice Unit Test Free Response Unit Test AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 7 of 16

8 Unit Frameworks Unit of Study: Applications of The Derivative Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural College Board Goal Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should use derivatives to solve a variety of problems. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to solve problems, experiment, interpret results, and verify conclusions. Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. 3. Applications of Derivatives 3.1. Analysis of functions Increasing/decreasing, concavity Graphically as rates of change Analytically as signs of derivatives Relative extrema First and Second Derivative Tests 3.2. Absolute extrema and optimization problems 3.3. Newton s Method 3.4. Mean Value Theorem and Average Value of Change Notes on Unit 3 This introduces derivatives as a tool for analysis of the behavior of a function. Analysis and justification of conclusions becomes more prominent with more emphasis on cogent written arguments, beyond simple mathematical justification. Newton s method introduces the concept of an iterative numerical process to solve an algebraic problem. AP Calculus will devote significant time to the theoretical development of concepts. Calculus and Calculus AP Frameworks.doc Page 8 of 16

9 Assessments Written Assignments Quizzes Multiple Choice Unit Test Free Response Unit Test Other Evidence AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 9 of 16

10 Unit Frameworks Unit of Study: The Indefinite Integral Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural College Board Goal Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should use derivatives to solve a variety of problems. Students should understand the relationship between the derivative and the definite integral as expressed by both parts of the Fundamental Theorem of Calculus. Interim Unit 3i The Indefinite Integral Techniques of integration anti-differentiation by substitution Families of antiderivatives Notes on Unit 3i This follows the winter break and is prior to the semester exam. Emphasis is on the mechanics of anti-differentiation. Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 10 of 16

11 Unit Frameworks Unit of Study: The Definite Integral Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit College Board Goal Students should understand the meaning of the definite integral both as a limit of Reimann sums and a net accumulator of change and should use integrals to solve a variety of problems. Students should understand the relationship between the derivative and the definite integral as expressed by both parts of the Fundamental Theorem of Calculus. Students should be able to communicate mathematics both orally and in well written sentences and should be able to explain solutions to problems. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to solve problems, experiment, interpret results, and verify conclusions. Objectives o Conceptual o Factual o Procedural Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. 4. The Definite Integral 4.1. Area under a curve Sigma notation Area as a limit Reimann sums 4.2. Definition of the Definite Integral Associated definitions and theorems 4.3. Fundamental Theorems of Calculus 4.4. Evaluation of Definite Integrals By First Fundamental Theorem By substitution By calculator Use of the calculator to verify analytical indefinite integration 4.5. Application of the Definite Integral as an infinite accumulator Calculus and Calculus AP Frameworks.doc Page 11 of 16

12 Geometric Area between curves Volumes of solids of revolution Volume by parallel section Arc length and surface area 4.6. Mean Value Theorem & Average Value of a Function 4.7. Rectilinear Motion II Notes on Unit 4 Definite integrals are introduced as area under curves and as a limit process. The emphasis on applications is using the definite integral as an accumulator of infinitesimal changes. Much of the applications make use of the numerical integration capacity of the calculator. Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes Multiple Choice Unit Test Free Response Unit Test AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 12 of 16

13 Unit of Study: Unit Frameworks Transcedental Functions and Differential Equations Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit College Board Goal Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations. Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should use derivatives to solve a variety of problems. Students should understand the meaning of the definite integral both as a limit of Reimann sums and a net accumulator of change and should use integrals to solve a variety of problems. Students should understand the relationship between the derivative and the definite integral as expressed by both parts of the Fundamental Theorem of Calculus. Students should be able to communicate mathematics both orally and in well written sentences and should be able to explain solutions to problems. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to solve problems, experiment, interpret results, and verify conclusions. Objectives o Conceptual o Factual o Procedural Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. 5. Transcendental Functions & Differential Equations 5.1. Inverse functions Derivatives of inverse functions 5.2. Review of properties of exponentials and logarithms 5.3. The Natural Logarithm as a Definite Integral Properties Graph Calculus and Calculus AP Frameworks.doc Page 13 of 16

14 Inverse Properties and graph Evaluation of e as a limit 5.4. The calculus of logarithms and exponentials Derivation of the rules Logarithmic differentiation 5.5. Inverse Trigonometric Functions Review of inverse trigonometric functions Domain and ranges Evaluation of composites Derivatives of inverse trigonometric functions Integrals yielding inverse trigonometric functions 5.6. Integration by Parts (AP) 5.7. Differential Equations Simple Separable variables Applications Slope fields (AP) Notes on Unit 5 Natural logarithms are presented as integrals. This reinforces the concept of functions defined from integrals and the use of the second fundamental theorem. Although not part of the AB curriculum, integration by parts is presented as a gateway to a variety of integration techniques (AP). Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes Multiple Choice Unit Test Free Response Unit Test AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 14 of 16

15 Unit Frameworks Unit of Study: Assorted Topics and Exam Review Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural College Board Goal Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment. Interim Unit 5i Assorted Topics and Exam Review L Hôpital s Rule Trapezoidal Rule (AP) Notes on Unit 5i Emphasis is on preparation for the AP exam (AP). Sample exams are given and strategies are presented (AP). Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 15 of 16

16 Unit of Study: Techniques of Integration (AP Only) Unit Frameworks Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural College Board Goal Students should understand the relationship between the derivative and the definite integral as expressed by both parts of the Fundamental Theorem of Calculus. 6. Techniques of Integration 6.1. Integrals of powers and products of trigonometric functions (AP) 6.2. Integration by trigonometric substitution (AP) 6.3. Integration of rational functions using partial fractions (AP) 6.4. Integrals tables (AP) Notes on Unit 6 This is after the AP exam when motivation lags and short simple mechanical processes can be presented and learned (AP). The unit test is a multi-day take-home multiple-choice test that is comprehensive over the entire course (AP). Assessments AP Calculus will devote significant time to the theoretical development of concepts. Written Assignments Other Evidence Quizzes Multiple Choice Unit Test Free Response Unit Test AP Calculus AP AB Calculus Exam Calculus and Calculus AP Frameworks.doc Page 16 of 16

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only.

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only. Calculus AB Priority Keys Aligned with Nevada Standards MA I MI L S MA represents a Major content area. Any concept labeled MA is something of central importance to the entire class/curriculum; it is a